Diagnosing On-Leveling Calculation Errors

Why On-Leveling Errors Matter

On-leveling adjusts historical earned premium to the rate level that applies at a selected target date. A calculation can use an appropriate method and still be wrong if it assigns a rate change to the wrong date, weights the rate levels incorrectly, reverses a factor, or combines sequential changes additively. These errors affect the premium basis used in later ratemaking calculations.

A useful diagnostic is to ask what the adjusted premium represents: the premium that would have been earned on the same exposure at the target rate level. Every factor and weight should be consistent with that interpretation. The checks in this tutorial are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $R_i$ be the rate level applicable to a portion of historical earned premium $P_i$, and let $R_c$ be the target rate level. If the exposure and coverage basis is unchanged, the on-level factor for that portion is $R_c/R_i$, and its on-leveled premium is $P_i(R_c/R_i)$. The factor is target rate level divided by historical rate level.

Rate levels may be represented by indices, such as 100, 110, and 104.5. Their units are arbitrary, but their ratios must represent the relative rates. If the historical premium is already stated at the applicable historical rate level, multiply it by the factor; do not divide by the factor.

A Reliable Diagnostic Sequence

Before calculating, identify the date that controls each rate change and establish which rate level applies to each portion of earned premium. Then check the earning weights and the direction of every factor. Finally, calculate the adjusted premium both by applying factors to the historical premium and by comparing the corresponding exposure-weighted rate levels, where that comparison is available.

Use the effective date.
A rate change applies from its stated effective date, not automatically from its approval, filing, or announcement date. Determine which policies or exposures are subject to the changed rate.

Establish the earning weights.
Identify the fraction of earned exposure, or the earned premium amount, at each rate level. Match the proportions to the specified policy term, experience period, and earning assumptions.

Orient each factor.
For historical premium at level $R_i$, use $R_c/R_i$. A historical rate below the target level should produce a factor above 1; a historical rate above the target should produce a factor below 1.

Compound sequential changes.
Apply each percentage change to the rate level that exists immediately before it. Do not add percentage changes unless the problem explicitly defines changes in additive index points.

Why the Factor Has This Direction

Suppose a group has $E_i$ units of exposure at historical rate level $R_i$. Its historical premium is $P_i=E_iR_i$, up to a common conversion scale. At the target level, the same exposure would produce $E_iR_c$. Substituting $E_i=P_i/R_i$ gives the adjusted premium:

E_iR_c=\frac{P_i}{R_i}R_c=P_i\frac{R_c}{R_i}.

Thus the factor is current divided by historical. If the historical level is 100 and the target is 120, the same exposure should generate 20% more premium, so the factor must be 1.20. The reversed factor, $100/120$, would instead reduce the premium and cannot represent the target adjustment.

When portions have different levels, adjust each portion and then sum. Equivalently, if weights $w_i$ are fractions of exposure and the rate level is proportional to premium per exposure, the exposure-weighted historical level is $\sum_i w_iR_i$, and the factor to target level is $R_c/\sum_i w_iR_i$. If weights are shares of historical earned premium rather than exposure, use the premium-weighted average of the individual factors instead. Mixing these two kinds of weights is a frequent source of error.

Worked Example: An Effective Date Is Not an Approval Date

A hypothetical insurer has an old rate level of 100. A change to level 110 is approved on June 15 but is effective for policies beginning on or after July 1. The target level is 121. Historical earned premiums are $500 for policies effective June 20, $1,000 for policies effective July 1, and $500 for policies effective October 1. Calculate the on-leveled premium. Assume each listed premium was earned at the rate level applicable to that policy’s effective date and that coverage and exposure basis are unchanged.

Method: apply the effective-date rate level to each premium portion. The June 20 policies begin before July 1, so they remain at level 100. The July 1 and October 1 policies are at level 110. The June 15 approval date does not make the change effective for the June 20 policies.

For the $500 at level 100, the factor is $121/100=1.21$, giving $500(1.21)=\$605$. For the $1,000 at level 110, the factor is $121/110=1.10$, giving $1{,}000(1.10)=\$1{,}100$. For the $500 at level 110, the adjusted premium is $500(1.10)=\$550$.

P_{\text{on-level}}=605+1{,}100+550=\$2{,}255.

The historical premium is $500+1{,}000+500=\$2{,}000$, so the aggregate factor is $2{,}255/2{,}000=1.1275$. The check is that each portion was increased by the ratio of target level to its actual historical level. Common mistake: treating June 15 as the effective date would assign the June 20 portion to level 110 and adjust it to $550 rather than $605, understating the result by $55.

Check the Earning Proportion Against the Period

A proportion is not correct merely because it looks plausible. It must correspond to the period whose earned premium is being adjusted and to the assumptions about policy effective dates, policy term, and earning pattern. For example, a rate change effective late in a calendar year does not ordinarily mean that the same fraction of that year’s earned premium is at the new level: policies earn over time, and policies subject to the change may take time to contribute earned premium.

Under the standard parallelogram assumptions—uniform policy effective dates, annual policies, a change applying at new and renewal effective dates, and uniform exposure earning—the fraction of a calendar year’s earned exposure at the new level for a change effective $m$ months before year-end is $\tfrac12(m/12)^2$. This is not a universal proportion: it does not apply unchanged when the policy term, effective-date distribution, or earning pattern differs. The key diagnostic is to use a proportion derived for the stated setup, not to substitute the fraction of calendar months after the change.

Worked Example: Correcting a Calendar-Month Proportion

A hypothetical portfolio has annual policies with uniform effective dates and uniform earning. A rate change applies at new and renewal effective dates on October 1. What fraction of earned exposure during that calendar year is at the new level? Compare it with the mistaken calendar-month calculation.

Method: use the parallelogram proportion under the stated assumptions. There are three months from October 1 to year-end, so $m=3$. The new-level fraction is:

\frac12\left(\frac{m}{12}\right)^2 =\frac12\left(\frac{3}{12}\right)^2 =\frac12\left(\frac14\right)^2 =\frac{1}{32}=0.03125.

Therefore, 3.125% of the calendar year’s earned exposure is at the new level, and the complementary old-level fraction is $1-0.03125=0.96875$, or 96.875%. The fractions sum to 100%, as required.

Common mistake: using $3/12=25\%$ as the new-level earned proportion. That counts every exposure in the last three calendar months as if it were already subject to the new level. Under the stated annual-policy assumptions, the share at the new rate builds up during those months rather than appearing all at once. The parallelogram result, not the calendar-month share, is appropriate here.

Sequential Rate Changes Must Be Compounded

A percentage change applies to the rate level in force immediately before that change. For example, a 10% increase followed by a 5% decrease gives $R_0(1.10)(0.95)$, not $R_0(1+0.10-0.05)$. The second change is 5% of the level after the increase. When building a rate-level timeline, record the level after each effective change before assigning factors to premium portions.

Worked Example: Fixing Additive Sequential Changes and a Reversed Factor

A hypothetical product has rate level 100 initially, increases by 10%, and later decreases by 5%. The current target is the level after both changes. Historical earned premium is $20,000 at the initial level, $30,000 after the increase but before the decrease, and $50,000 after both changes. Calculate the target level and on-leveled premium.

Method: compound the rate changes, then adjust each premium portion. The level after the 10% increase is $100(1.10)=110$. The subsequent 5% decrease gives $110(0.95)=104.5$, the target level.

The $20,000 at level 100 is adjusted by $104.5/100=1.045$, giving $20,900. The $30,000 at level 110 is adjusted by $104.5/110=0.95$, giving $28,500. The $50,000 already at level 104.5 has factor $104.5/104.5=1$, giving $50,000.

P_{\text{on-level}}=20{,}900+28{,}500+50{,}000=\$99{,}400.

The original premium total is $100,000, so the aggregate factor is $99{,}400/100{,}000=0.994$. This is plausible: some premium is at a level above the target, so the adjustment can reduce the total. Common mistakes: adding the changes to get level 105, or reversing the factor for the level-100 premium to $100/104.5$. The first ignores compounding; the second would reduce the premium at the lowest historical level, contrary to the target adjustment.

Common Diagnostic Checks

Check the date label. Distinguish a change’s approval or announcement date from its effective date, and determine which policy transactions are actually affected.
Check the denominator. For historical premium at level $R_i$, the denominator is $R_i$, not the target level $R_c$.
Check the weights. Do not treat calendar months, earned-exposure proportions, and earned-premium shares as interchangeable.
Check the rate-level sequence. Write the level after each change; apply each percentage to the level immediately before it.
Check the result’s direction. Premium from a level below target should increase; premium from a level above target should decrease.

These checks do not replace the selected on-leveling method. They help reveal whether the dates, proportions, and arithmetic used by that method agree with the stated assumptions. If those assumptions do not support a particular proportion—for example, because effective dates are not uniform—do not force a standard proportion onto the calculation.

Key takeaway: An on-level factor converts historical premium to the target rate level, so its direction is target divided by historical. Assign rate levels using effective dates, use earning proportions that match the stated exposure pattern, and compound sequential percentage changes.

Check Your Understanding

For each problem, identify the relevant rate levels and check the factor direction before calculating.

A portfolio has 40% of its earned exposure at level 100 and 60% at level 125. The target is level 125. Assuming the shares are exposure weights and the rate level is proportional to premium per exposure, find the aggregate on-level factor. (Answer: $125/(0.40(100)+0.60(125))=1.0870$)
A change from level 80 to level 88 is approved February 10 but effective March 1. Historical earned premiums are $800 for policies effective February 20 and $1,200 for policies effective March 1. The target is level 96.8. Find the on-leveled premium. (Answer: $800(96.8/80)+1{,}200(96.8/88)=\$2{,}288$)
A rate level of 100 increases by 8%, then increases by 5%. What is the resulting level? (Answer: $100(1.08)(1.05)=113.4$)
Historical earned premium is $10,000 at level 100 and $20,000 at level 125. The target is level 125. Find the total on-leveled premium and aggregate factor. (Answer: $$12{,}500+$20{,}000=\$32{,}500$; factor $32{,}500/30{,}000=1.0833$)
Under uniform effective dates, annual policies, uniform earning, and a change applying at new and renewal effective dates, a change is effective October 1. What fraction of that calendar year’s earned exposure is at the new level? (Answer: $\tfrac12(3/12)^2=0.03125$, or 3.125%)

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